continuum hypothesis

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If you are looking for an answer to the question What is Artificial Intelligence? and you only have a minute, then here's the definition the Association for the Advancement of Artificial Intelligence offers on its home page: "the scientific understanding of the mechanisms underlying thought and intelligent behavior and their embodiment in machines."

However, if you are fortunate enough to have more than a minute, then please get ready to embark upon an exciting journey exploring AI (but beware, it could last a lifetime) …

Artificial intelligence (AI) is trending globally in commerce, science, health care, geopolitics, and more areas. Deep learning, a subset of machine learning, is the lever that launched the worldwide rush--an area of strategic interest for researchers, scientists, visionary CEOs, academics, geopolitical think tanks, pioneering entrepreneurs, astute venture capitalists, strategy consultants, and management executives from companies of all sizes. Yet in the midst of this AI renaissance, is a relatively fundamental unsolvable problem with machine learning that is not commonly known, nor frequently discussed outside of the small cadre of philosophers, and artificial intelligence experts. A global research team of researchers have recently demonstrated that machine learning has an unsolvable problem, and published their findings in Nature Machine Intelligence in January 2019. Researchers from Princeton University, the University of Waterloo, Technion-IIT, Tel Aviv University, and the Institute of Mathematics of the Academy of Sciences of the Czech Republic, proved that AI learnability cannot be proved nor refuted when using the standard axioms of mathematics.

In the late nineteenth century, Georg Cantor, the founder of set theory, demonstrated that not all infinite sets are created equal. In particular, the set of integer numbers is'smaller' than the set of all real numbers, also known as the continuum. Cantor also suggested that there cannot be sets of intermediate size that is, larger than the integers but smaller than the continuum. According to Continuum hypothesis, no set of distinct objects has a size larger than that of the integers but smaller than that of the real numbers, the statement, which can be neither proved nor refuted using the standard axioms of mathematics. In the 1960s, US mathematician Paul Cohen showed that the continuum hypothesis cannot be proved either true or false starting from the standard axioms the statements taken to be true of the theory of sets, which are commonly taken as the foundation for all of mathematics.

In a study published in Nature Machine Intelligence, researchers discovered that in some cases of machine learning it cannot be proved whether the system actually'learned' something or solved the problem. They explore machine learning learnability. We already know that machine learning systems, and AI systems in general are black boxes. You feed the system some data, you get some output or a trained system that performs some tasks but you don't know how the system arrived at a particular solution. Now we have a published study from Ben-Davis et al that shows learnability in machine learning is undecidable.

In a world where it seems like artificial intelligence and machine learning can figure out just about anything, that might seem like heresy – but it's true. At least, that's the case according to a new international study by a team of mathematicians and AI researchers, who discovered that despite the seemingly boundless potential of machine learning, even the cleverest algorithms are nonetheless bound by the constraints of mathematics. "The advantages of mathematics, however, sometimes come with a cost… in a nutshell… not everything is provable," the researchers, led by first author and computer scientist Shai Ben-David from the University of Waterloo, write in their paper. "Here we show that machine learning shares this fate." Awareness of these mathematical limitations is often tied to the famous Austrian mathematician Kurt Gödel, who developed in the 1930s what are known as the incompleteness theorems – two propositions suggesting that not all mathematical questions can actually be solved.

Mathematicians have discovered a problem they cannot solve. It's not that they're not smart enough; there simply is no answer. The problem has to do with machine learning -- the type of artificial-intelligence models some computers use to "learn" how to do a specific task. When Facebook or Google recognizes a photo of you and suggests that you tag yourself, it's using machine learning. Neuroscientists use machine learning to "read" someone's thoughts.

Austrian mathematician Kurt Gödel is known for his'incompleteness' theorems.Credit: Alfred Eisenstaedt/ LIFE Picture Coll./Getty A team of researchers has stumbled on a question that is mathematically unanswerable because it is linked to logical paradoxes discovered by Austrian mathematician Kurt Gödel in the 1930s that can't be solved using standard mathematics. The mathematicians, who were working on a machine-learning problem, show that the question of'learnability' -- whether an algorithm can extract a pattern from limited data -- is linked to a paradox known as the continuum hypothesis. Gödel showed that the statement cannot be proved either true or false using standard mathematical language. The latest result appeared on 7 January in Nature Machine Intelligence1.

During the twentieth century, discoveries in mathematical logic revolutionized our understanding of the very foundations of mathematics. In 1931, the logician Kurt Gödel showed that, in any system of axioms that is expressive enough to model arithmetic, some true statements will be unprovable1. And in the following decades, it was demonstrated that the continuum hypothesis -- which states that no set of distinct objects has a size larger than that of the integers but smaller than that of the real numbers -- can be neither proved nor refuted using the standard axioms of mathematics2–4. They identify a machine-learning problem whose fate depends on the continuum hypothesis, leaving its resolution forever beyond reach. Machine learning is concerned with the design and analysis of algorithms that can learn and improve their performance as they are exposed to data.